Publications of the Department of Astronomy -
Beograd, N2 10, 1980
UDC 523.24; 521.1/3
OSP
THE RADIATION PRESSURE AND THE MOTION OF THE
ARTIFICIAL SATELLITES
L. Sehnal
Astronomical Institute of the Czechoslovak Academy of Sciences,
Observatory Ondtejov
Received January 30, 1980
Summary. The paper deals with the radiative effects on the motion of the Earth artificial
satellites. Separately, the solar and terrestrial radiation sources are taken into account and the
Poynting-Robertson effect is treated, too. The magnitude of the disturbing forces is compared
to that of the atmospheric drag and relevant effective heights are computed. The components of
the disturbing forces are determined and the changes of the elements are obtained from the planetary Lagrange-Gauss equations.
Ladislav Selmal, PRITISAK ZRACENJA I KRETANJE VESTACKIH SATELlTAOvaj rad uzima u obzir uticaj zracenja na kretanje ZemIjinih V'estaCkih satelita. Uzeti su u razmatranje ponaosob suncano i zema1jsko zracenje, a takode je razmatran i Poynting-Robertsonov efekt.
Velicina poremeeajnih sila uporedena je sa velicinom atmosferskog otpora i izraeunate su odgovarajuCe efekUvne visine. Odredene su komponente poremeeajnih sila i dobijeDe promene elemeData iz planetskih Lagran1-Gausovih jednacina.
For most satellites, at least for those which come close to the Earth, the atmospherical drag is the most important non-graitational disturbing effect. Next to
this is the radiation pressure which can outgrow the drag in greater distances from
the Earth.
. Let us just quote briefly the different kinds of radiation effects:
1) Direct solar radiation pressure
2) Earth albedo radiation pressure
3) specularly reflected radiation pressure
4) infrared radiation pressure
5) Poynting-Robertson effect.
[ All those effects, like drag, depend on the satellite effective area/mass ratio.
Close to the Earth, the influence of the atmosphere prevails; with growing height
the density of the atmosphere rapidly diminishes and the drag effect declines, too.
Making a ratio of all the above quoted effects to drag, the satellite area/mass ratio
15
Ladislav SehnaJ, The radiation pressure and the motion of the artificial satellites
disappears so that for all satellites, we can find a certain height, for which the radiation pressure effects become greater than drag.
We can write for the already mentioned disturbing accelerations the approximate formulas under some simplifying conditions:
a) Drag:
b) Direct solar radiation pressure:
A
Fs=K-Q
m
c) Earth albedo radiation pressure:
FA=~ K
4
A Qct(R)2
m
r
d) Infrared radiation pressure:
FIR=K A
m
Ao(R)2 +J.. K A A2(R)2
r
m
4
r
e) Poynting-Robertson effect:
A
V
FPR=-Qm
c
We omit in our analysis the pressure of the specularly reflected radiation
(from the Earth surface) as a very random effect (depending strongly on ocean-land
difference, cloud cover etc.).
The infrared radiation pressure has two components - the first one comes
from the homogeneous spherically symetrical component which actually does not
contribute to any perturbations of the elements. As we shall see later, it acts only
like diminishing the effective gravity. The disturbing component originates from
the second term with A2. The constants Ao, A2, etc. are the constants in the spherical harmonic representation of the terrestrial infrared radiative field (1.
Making the ratio of the radiative disturbing forces to drag equal to unity,
we have the conditions for determination the radius-vector or height above which
the radiative forces prevail:
a) Direct solar radiation pressure:
p/r=~.JL
2 GM
b) Albedo radiation pressure:
1 QctR2
rp=---
4 GM
c) Infrared radiation pressure:
1 AoR2
2 GM
rp=---
16
Ladislav School, The radiation pressure and the motion of the artificial satellites
d) Disturbing infrared radiation pressure:
1 A2R2
rp=---
8 GM
e) Poynting-Robcrtson effect:
plr=~ --.!L~.
2 GM c
On the left side, we have the terms depending on the radius-vector (or height) and
if we make choice of a specific model of the atmosphere we can solve those expressions for r. However, since the density depends strongly on the solar activity, we
shall have accordingly different values of r. Fig. 1 shows such dependence; the
upper regions indicate the heights above which the radiative effects prevail.
h
(km)
OL-----~----
O'
__
~
____
1500'
~
______
2000'
L __ __
2500' T co
Fig. 1. The heights at which th(' different radiative effects exceed th atmospherical drag, in terms
of growing solar activity represented by exospheric temperature Too.
In the following, we shall discuss in some more detail the individual radiative effects.
To compute the perturbations caused by the direct solar radiation pressure
we can use the equation for F s, where the constant Q means the pressure exerted
on a unit surface in 1 astr. unite distance from the Sun, and it is Q = 4.63 X 10-5
in cgs system.
2
17
Ladislav Sehnal, The radiation pressure and the motion of the artificial satellites
The computation is not complicated in case the satellite does not enter the
Earth shadow. We suppose that the direction to the Sun does not change and the
disturbing function can be then written as
Fs= -Q A (x cosLs+y cos e:cosLs+zsine:sinLs),
m
where x, y, z are the geocentric coordinates of the satellite, e; and Ls is the ec1iptical
obliquity and longitude of the Sun (1).
Immediately after introducing the expression for F s into the equations for
variations of constants, we see that there are no secular perturbations of the semi-major axis. However, this does not hold in case the satellite enters the Earth shadow during its revolution around the Earth. In this case we can, of course, integrate the equations within limits corresponding to the entry and exit of the satellite into or from the shadow.
Another method which could enable to integrate over whole revolution and
so it would be independent on the conditions of the shadow entry, lies in introducing the so called "shadow function". This function should gain a value of unity
on the illuminated portion of the orbit and drop to zero during the shadow crossing. Multiplying the disturbik function by this "shadow" function, we have actually the exact description of the conditions of a real satellite orbit.
Basic problem in construction of the shadow function lies in the mathematical approximation of a discontinuous function. The geometrical situation can be
shown on following Fig. 2:
-
Sun
Fig. 2. The shadow function x.
The angular distance A of the satellite from the center of the shadow can be expressed as
cos A = Al cos 'D + Aa sin 'D,
where the coefficients Al and Aa depend on the longitude of the Sun Ls, obliquity
of the ecliptic and on the orbital elements. Similarly, the angle <I> can be written as
sin <p = eRlp)(I + ecos'D).
If we write the difference A - <p = x, then the function
x=J... (1 + xlix /) =J... [1 +sinxl (I_COS 2 x)1/2]
2
2
has the properties we are looking for (2).
18
Ladi!lav Sfhnal, The radiation pressure and the motion of the artificial satellites
This function K can be developed into a power series where the argument
is the true anomaly v. For US2.ge of this function in the Lagrange equations we transform the true anomaly into the mean on~ using the Hansen coefficients and then
we have for the ch2.nge of an arbitrary clement C1 the expression
6cr=FD[SM -+-
EPj sinjM -+- E Qj (l-cosjM)]
j
j
This is exactly the same form as wc got for the change of arbitrary element cr caused by drag; since the coefficients Sj, Pj ~!fld Qj are given again by multiplication
of several series, we shall use the semi~analytical method, consisting of expressing
the coefiicients as the functions of indeces or as reccurent formulas.
The shadow function can be introduced through several other mathematical
means, e. g. the Fourier series (3) or Legendre polynomials (4). It seems, however,
that the power series give the fastest convergence.
The analytical description of the albedo effect is very difficult even when we
take into occount the simplest model of the albedo - a constant value, homogeneously distributed over the Earth's surface. In this si:nple case it is possible to
find an analytical expression which agrees with the numerical model in its limiting
values (vanishes when satellite enters the Earth shadow). Such a function can be
expressed, e. g. as
FARP = FARPc cos 2 Tj (7t/2 + cos-1R/r) 7t/2
where Tj is the angle between the satellite radius-v.xtor and the Earth-Sun line.
The quantity Fc is the maximum value of the albedo radiation pressure which
can be computed analyticaly in a close form (5).
In last years, there were attempts to find the albedo effects using some special methods. In this sense we have to mention the french satellite D-5-B having
on board the microaccelerometer with which it was possible to measure even small
accelerations effecting the motion of the satellite. When the satellite was illuminated ba the Sun, the direct solar radiation pressure, infr.1rI~d radiation pressure and
the albedo radiation was active. Since the direct radiation has a constant character,
as well as the infrared, the changes of the acceleration could have been assigned
to the albedo radiation. This was in those part of the orbit, when the drag effects
were small owing to great distance from the Earth.
The measurents made by the microacceleromet·:r showed clearly the distinction between land and oceans and also the latitudinal dependence. Thus,
the albedo can be represented by two formulas (6):
a. = 0.1 + 0.3 sin I q:ll
for ocean,
a. = 0.2 + 0.3 sin I q:ll
for land,
where q:l is the latitude.
This is a description of the albedo distribution by discrete values; however,
by Fourler analysis one can get a continuous model. Since the D-5-B satellite's 30°
inclination, the higher latitudes were not covered by the microaccelerometer measurements, so that the snow covered arctic areas got smaller albedo values. We
tried to make a model of the Earth albedo using spherical harmonic analysis and
taking into account the surface effects. The c('mparison (If the longitudinal averages with the values made by the satellite Tiros 7 show good agreement (7), see
Figs. 3, 4.
2*
19
Ladislav Sehnal, The radiation pressure and the motion of the artificial satellites
ALBEDO 12,12 Summer
50
40
30
20
45
35
25
15
15
25
20
30
40
50
35
45
900W
i800W
900 E
0°
1800 E
Fig. 3. The summer Earth albedo model 12, 12. The values of the albedo isolines are in percents.
1.0
A
I-
0.8
I-
SUMMER ALBEDO
I-
0.5 r-
0
TIROS 7
000
.00
!t •
I--
~
0.4
I-
..
0
o
o
9
0
0
•
MODEL 12,12
0
o
0
o
0
0
0
0
o
I-
0.2
0
0
0
o
o
0
0
"
l-
11
o
0
Q
0
0
0
0
o
0
~ ~
0
0
0
0
0
o
0
•
I-
0.0
+90
I
+50
I
I
I
+30
I
I
0
I
I
I
-30
I
I
I
-SO
I
I
'f
-90
Fig. 4. The longitudinal averages of the albedo in 5deglatitudinal (<p) zones as measured by Tiros 7
and as given by our summer albedo model.
The terrestrial infrared radiation pressure is very difficult to register, since
its main part, which comes from the homogeneously radiating spherical surface
does not make any changes of the elements. To treat the infrared radiation more
precisely, let us represent it by a spherical harmonic expansion (8), where we shal
take into account the zonal terms of zero and second degree (9):
er = Ao + A2 P2 (sin cp),
20
Ladislav Sehnal. The radiation pressure and the motion of the artificial satellites
where P2 is the Legendre polynomial of second degree. The geometry of the effect
is on Fig. 5; we shall suppose the Lambert law is valid for the outgoing radiation,
so that the acceleraton dFJR exerted by the infrared radiation flux from an area
dP on a satellite of a unit arealmass ratio and of a unit surface reflectivity will be
given as
dFJR = 1/7t ale I/p2 cos Z dP.
dS
dF
Fig. 5. The geometry of the terrestrial infrared radiation pressure dF (and its radial component
dS) from the infinitessimal area dP.
The integration of dF over the whole cap visible to the satellite can be performed
in a closed form, however, after introducing those expressions into the equations
for variations of constants we would not be able to integrate them. Therefore we
shall use the development using the Gegenbauer (ultraspheric) polynomials and
shall have the result e. g. for S:
. tpB)
S= Aoy-2 + A2 P 2 (sm
y-2 [ - I
+ -4
4
5
y-l
+ -I
4
18 y-3 _ _4_ y-5]
y-2 _ _
35
105
The expressions for the other terms have similar form, exept the terms with Ao
vanish, as we could expect.
To integrate the equations for the variations of constants, we shall introduce
the mean anomaly and get for the changes after one revolution the integrals of the
form:
(: r
cos mv dM,
where v is the true and M the mean anomalies. We can use e. g. Tisserand's treatise (10) to solve the integrals and finally we shall come to the changes for the elements in form of series in powers of (Rlp)fI, where R is the- equatorial radius of the
Earth and p is the orbital parameter.
21
Ladislav Sehnal. The radiation pressure and the motion of the artificial satellites
To check the results numerically, Wo:' could use only the measurements made
by the microaccelerometer on board of D-S-B. All measured data were assigned
to the effect of the radial component S. The numerical results computed from our
theory are in complete agreement (S = 3.S37 X 10-9 m/sec2). The statistics of the
observed data showed a slight growth of the infrared radiation effect from equatorial to tropical regions. This would contradict to the results of our theory but let
us remind that the measurements give rather statistical estimate of the effect and
on the other side, the theory includes just one disturbing term (second zonal harmonics).
The changes of the elements were not yet observed, but let us give at least
predicted values (per day):
D-S-B
!1a
!1 Tp
!1e
!1i
!1'o
!16)
Lageos
3.7 cm
- l2.S cm
2.2x 10-8
7.6 x 10-8 deg
- S.OX 10-5 deg
- 1.4 x 10-5 deg
O.OOOS cm
- O.OScm
3.8 x 10-11
- 9.6x 10-13 deg
- 2.Sx 10-7 deg
- 6.9x 10-8 deg
REFERENCES
1. Kozai, Y.
1961, Smith. Astroph. Ohs. Special Report, No. 56.
1969, The Earth shadowing effects in the short-periodic perturbations
2. Lala, P., Sehnal, L.
of satellite orbits, Bull. Astr. Inst. Czech., Vol. 20, 327-330.
3. Ferraz-Mello, S.
1964, Sur la probleme de la pression de la radiation dans la theorie des
satellites artificiels, C. R. Acad. Sc. Paris 25S, 463-466.
4. Vashkoviak, S. N.
1973, Effects of radiation pressure on the motion of artificial satellite;
Ha6J1& HCKycciU6. CUyiUHUK06 3eM"u, No. 15, Moscow 1973, 124-143.
5. Sehnal, L.
1978, Spherical harmonic analysis of the Earth's albedo, Ha6J1& HCKycciU6.
CUyiUHUK06 3eMJlu, No. 18, Warszawa 1979, 315-322.
6. Lata, P., Barlier, F., Oyharcabal G.
1978, Interpretation of the D-5-B Satellite Measurements and the New Modle of the Earth's Alb:do, Bull. Altr. Inst. Czech., Vol. 29, 238243.
1979, Tb.! Earth alb!d1 mJdel in spherical h!lrmJnics, Bull. Astr. Inst. Czech.,
7. Sehnal, L.
Vol. 30, 199-204.
8. Manakov, Tu., M., Pellinen, L., N.
1979, The influence of the terrestrial radiation of the
Earth on the motion of Earth artificial satellites, KOCMU14eCKUe uCCJle006aHUil, 17,498-521.
9. Sehnal, L.
1980, The motion of an artificial satellite under the terrestrial infrared radition pressure, Cel. Mech.
10. Tisserand, F.
1889, Traite de mechanique celeste, T. I, 249-261, Gauthier-Villars, Paris.
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